Constraints on Planet Nine's Orbit and Sky Position within a Framework of Mean Motion Resonances
A number of authors have proposed that the statistically significant orbital alignment of the most distant Kuiper Belt Objects (KBOs) is evidence of an as-yet undetected planet in the outer solar system, now referred to colloquially a "Planet Nine". Dynamical simulations by Batygin & Brown (2016) have provided constraints on the range of the planet's possible orbits and sky locations. We extend these investigations by exploring the suggestion of Malhotra et al. (2016) that Planet Nine is in small integer ratio mean-motion resonances (MMRs) with several of the most distant KBOs. We show that the observed KBO semi-major axes present a set of commensurabilities with an unseen planet at $\sim 654 \ \mathrm{AU}$ ($P \sim 16,725 \ \mathrm{yr}$) that has a greater than $98\%$ chance of stemming from a sequence of MMRs rather than from a random distribution. We describe and implement a Monte-Carlo optimization scheme that drives billion-year dynamical integrations of the outer solar system to pinpoint the orbital properties of perturbers that are capable of maintaining the KBOs' apsidal alignment. This optimization exercise suggests that the unseen planet is most consistently represented with mass, $m \sim 6-12 M_{\oplus}$, semi-major axis, $a \sim 654 \ \mathrm{AU}$, eccentricity, $e\sim0.45$, inclination, $i \sim 30^{\circ}$, argument of periastron, $ω\sim 150^{\circ}$, longitude of ascending node, $Ω\sim 50^{\circ}$, and mean anomaly, $M \sim 180^{\circ}$. A range of sky locations relative to this fiducial ephemeris are possible. We find that the region $30^{\circ}\, \lesssim \mathrm{RA} \lesssim 50^{\circ}$, $-20^{\circ}\, \lesssim \mathrm{Dec} \lesssim 20^{\circ}$ is promising.